lean_workbook_plus_67710

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Given a natural number $n \geq 3$ . Prove that there exists a set $A\left(n\right)$ such that:\n1) The set $A\left(n\right)$ consists of $n$ distinct natural numbers.\n2) For any element $a\in A\left(n\right)$ , the product of all the other elements of $A\left(n\right)$ has remainder $1$ when divided by $a$ .\nIn other words, we want positive integers $a_1,\ldots a_n$ s.t. $\sum_{i=1}^n \frac 1{a_i}-\prod_{i=1}^n \frac 1{a_i}$ is an integer.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_67710 (n : ℕ) (hn : 3 ≤ n) :
    ∃ a : Fin n → ℕ,
      (∀ i : Fin n, 0 < a i ∧ ∀ j : Fin n, i ≠ j → a i ≠ a j) ∧
      ∀ i : Fin n, (∏ j : Fin n, if i = j then a i else 1) % a i = 1   :=  by sorry

The product includes only the selected entry instead of all the other entries, so its remainder is always0.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Use ∏_{j≠i}aⱼ (Lean: if i=j then1 else a j), with positive pairwise distinct entries, and require that product mod aᵢ=1 for every i.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.